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Statefinder diagnostic and constraints on the Palatini f(R) gravity theories

We focus on a series of $f(R)$ gravity theories in Palatini formalism to investigate the probabilities of producing the late-time acceleration for the flat Friedmann-Robertson-Walker (FRW) universe. We apply statefinder diagnostic to these cosmological models for chosen series of parameters to see if they distinguish from one another. The diagnostic involves the statefinder pair $\{r,s\}$, where $r$ is derived from the scale factor $a$ and its higher derivatives with respect to the cosmic time $t$, and $s$ is expressed by $r$ and the deceleration parameter $q$. In conclusion, we find that although two types of $f(R)$ theories: (i) $f(R) = R + αR^m - βR^{-n}$ and (ii) $f(R) = R + α\ln R - β$ can lead to late-time acceleration, their evolutionary trajectories in the $r-s$ and $r-q$ planes reveal different evolutionary properties, which certainly justify the merits of statefinder diagnostic. Additionally, we utilize the observational Hubble parameter data (OHD) to constrain these models of $f(R)$ gravity. As a result, except for $m=n=1/2$ of (i) case, $α=0$ of (i) case and (ii) case allow $Λ$CDM model to exist in 1$σ$ confidence region. After adopting statefinder diagnostic to the best-fit models, we find that all the best-fit models are capable of going through deceleration/acceleration transition stage with late-time acceleration epoch, and all these models turn to de-Sitter point ($\{r,s\}=\{1,0\}$) in the future. Also, the evolutionary differences between these models are distinct, especially in $r-s$ plane, which makes the statefinder diagnostic more reliable in discriminating cosmological models.